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BIEN425 – Lecture 10

BIEN425 – Lecture 10. By the end of the lecture, you should be able to: Describe the reason and remedy of DFT leakage Design and implement FIR filters using rectangular, Hanning, Hamming and Blackman windowing methods. DFT leakage.

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BIEN425 – Lecture 10

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  1. BIEN425 – Lecture 10 • By the end of the lecture, you should be able to: • Describe the reason and remedy of DFT leakage • Design and implement FIR filters using rectangular, Hanning, Hamming and Blackman windowing methods

  2. DFT leakage • Leakageoccurs because the DFT X(i) producesaccurate results only when input data has energy precisely at discrete analysis frequencies given by ifs/N. • What happens if input signal has component at intermediate frequencies? • This is due to correlation between two waves, one of which does not have an integral number of cycles inN points; therefore the sum for the correlation computation is not zero.

  3. Recall the convolution theorem, notice that when we sample we not only multiply by an impulse train but also bya rectangular window. • We previous stated that our discrete frequency spectrum is the convolution of an impulse trainwith the TRUE frequency spectrum of our signal • In reality, we are taking the convolution of an impulsetrain with the frequency spectrum of a rectangular function - the result of this is then convolved with theTRUE frequency spectrum of our signal.

  4. Why rectangular windows?

  5. Spectrum of rectangular window • For N points and window (unit value) length of K, we can obtain the frequency spectrum which takes the form of Dirichlet Kernel (Lecture10.m) Mainlobe First zero of the mainlobe occurs at n = N/K.

  6. When we do a DFT, we convolve the Dirichlet kernel with the impulse train. Instead of getting spikes for pure sinusoids, we get leakages. • To deal with this problem, wetypically use a windowwith a different Fourier transform than the rectangle. This is also known asapodization, which literally means “chop the feet off.” This expression refers to the reduction of themagnitude of sidelobes in the window frequency spectrum. • Matlab example

  7. Some windows

  8. Filter transfer function can be re-written as: • First we determine h(i) based on our filter specs, then we decide the windows w(i)

  9. Finding h(i) • Type 1 and Type 2: In general, given a m-th order linear phase filter exhibiting even-symmetric about i=m/2, with group delay t=mT/2

  10. Similarly for Type 3 and 4 linear phase filters: • To find h(i), simply insert the right form of Ar(f) based on the filter characteristics into the correct equation.

  11. In general, for Type 1 linear-phase filter with order m=2p, h(k) can be written as follows

  12. Example

  13. General strategy (ideal) • Pick m • Pick a window w(i) • Pick a Type 1 ideal impulse response h(i) from Table 6.1 • Compute bi = w(i)h(i) • Compute H(z)

  14. Example

  15. Comparing windows • For a general m-th order FIR low pass filter, we compare the transition bandwidth, passband ripple and stopband attenuation

  16. General strategy (non-ideal) • Pick m • Pick a window w(i) • Pick Ar(f) • Compute bi = w(i)h(i) based on your filter type (1-4) • For Type 1 and 2 • For Type 3 and 4 • Compute H(z)

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